Integrand size = 17, antiderivative size = 28 \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2038, 627} \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \]
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Rule 627
Rule 2038
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {-b-2 c x^2}{b^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {2 c \,x^{2}+b}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(27\) |
gosper | \(-\frac {x^{2} \left (c \,x^{2}+b \right ) \left (2 c \,x^{2}+b \right )}{b^{2} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\) | \(37\) |
default | \(-\frac {x^{2} \left (c \,x^{2}+b \right ) \left (2 c \,x^{2}+b \right )}{b^{2} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\) | \(37\) |
trager | \(-\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{\left (c \,x^{2}+b \right ) b^{2} x^{2}}\) | \(39\) |
risch | \(-\frac {c \,x^{2}+b}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {c \,x^{2}}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(49\) |
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {c x^{4} + b x^{2}} {\left (2 \, c x^{2} + b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \]
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\[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 \, c x^{2}}{\sqrt {c x^{4} + b x^{2}} b^{2}} - \frac {1}{\sqrt {c x^{4} + b x^{2}} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {c x}{\sqrt {c x^{2} + b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )} b \mathrm {sgn}\left (x\right )} \]
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Time = 13.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2\,c\,x^2+b}{b^2\,\sqrt {c\,x^4+b\,x^2}} \]
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